High repetition-rate, diode-pumped, solid-state (DPSS) Q-switched lasers with near-diffraction-limited TEM00 beams and high overall efficiency are used widely in scientific, security, sensing, and material processing applications. In all cases, it is desirable to have the laser output tailored to the application in terms of wavelength, pulse energy, pulse width, and repetition rate.
In materials processing, such important aspects as removal rates, kerf quality, and collateral damage can depend strongly on all of these variables. However, the degree of possible tailoring among these parameters is tightly constrained by well-understood underlying physics. The output pulse energy and repetition rate are limited by the pumping level and total extractable power available from the laser system, though techniques such as nonlinear frequency conversion can relatively efficiently transfer this power to other wavelengths.
The laser pulse width depends on the physical laser parameters (gain medium, cavity round trip time, etc.) and the initial inversion level, which determine the build-up time and the energy extraction dynamics, and so is typically strongly coupled to the pulse energy. This constraint becomes more problematic as one moves to higher pulse repetition frequency since the energy available per pulse is reduced, leading to longer build-up times, longer pulses, lower intensities, and ultimately lower efficiency in frequency conversion.
For many applications, however, it would be desirable to break this pulse energy-pulse width constraint, so that the pulse width could be selected independently of the pulse energy. If overall efficiency were also preserved, such a laser would be an extremely adaptable tool, making possible high-efficiency, frequency-converted lasers over a wide range of repetition rates and pulse widths.
This invention discloses a laser and method for operating it to achieve a stable output with variable pulse width and high efficiency over a wide range of repetition frequency.
Intra-cavity frequency converted lasers have the significant advantage of highly efficient conversion to other wavelengths, reduced peak and average intensities on nonlinear crystals, and reduced pulse-to-pulse noise levels compared to similar externally frequency converted lasers, but typically exhibit longer pulses than lasers with external frequency conversion.
This pulse lengthening is due to two effects: low linear losses and intensity clamping by the nonlinear coupling. Internally (intracavity) frequency-converted lasers typically have low linear losses to maximize frequency conversion efficiency. As the circulating intensity decays, the nonlinear losses decrease, so that the pulse decay stretches out. During this decay, energy is still extracted from the gain medium, but the instantaneous nonlinear efficiency is progressively falling, which reduces the overall conversion efficiency. The overall efficiency can be improved by increasing the nonlinear output coupling, but typically only at the cost of increasing the pulse width yet further, since the increased nonlinear coupling more effectively clamps the circulating intensity and energy extraction rate.
Lasers with controllable pulse width have been constructed using a variety of techniques. Among the earliest employed increased nonlinear optical (NLO) coupling to lengthen the pulse width as analyzed and demonstrated by Murray and Harris (J. E. Murray and S. E. Harris, “Pulse Lengthening via Overcoupled Internal Second-Harmonic Generation”, J. Appl. Phys. 41, pp 609-613, 1970; J. F. Young, J. E. Murray, R. B. Miles, and S. E. Harris, “Q-switched Laser with Controllable Pulse Length”, Appl. Phys. Lett. 18, pp. 129-130, 1971). They determined an optimal level of second harmonic coupling at which the maximum intensity at the harmonic frequency is achieved. For lower harmonic coupling levels, the pulse width is approximately constant. For harmonic coupling levels greater than the optimal level, the larger NLO coupling effectively clamps the circulating intensity and lengthens the pulse accordingly without a loss of efficiency. However, this technique can only produce pulse lengthening and offers only a relatively slow pulse adjustment mechanism since the nonlinear material temperature or angle must be tuned on a millisecond time scale.
Recent work has utilized self-doubling laser gain media to achieve similar effects (P. Dekker, J. M. Dawes, and J. A. Piper, “2.27-W Q-switched self-doubling Yb:YAB laser with controllable pulse length”, J. Opt. Soc. Am. B 22, pp. 278-384, 2005).
The invention disclosed herewith introduces an improvement over an earlier patent on Q-switch clipping of pulse falling edges (Adams, U.S. Pat. No. 6,654,391). The main objective of the Adams patent is to close the Q-switch after the majority of the frequency converted pulse is over in order to retain in the gain medium some fraction of the energy which would otherwise have been extracted by the trailing edge of the fundamental wavelength pulse.
More stored energy and gain are therefore available for the following pulse, resulting in higher intensity fundamental pulses, higher conversion efficiencies, and more power at the converted wavelength. Slight pulse shortening may also be achievable, but as the Adams patent states, as the Q-switch window becomes shorter and begins to clip significantly into the trailing edge of the pulse, the pulse width becomes unstable.
This instability typically takes the form of a period multiplication of the pulse train, so that instead of having a train of equal energy/intensity pulses, the pulses alternate between large and small energy, with the difference between them depending on the degree of trailing edge clipping by the Q-switch. Thus, the method of the Adams' patent cannot be used to achieve substantial pulse shortening.
Another theoretically possible technique for controlling the pulse width of a Q-switched laser is use of an aggressive Q-switch window terminating prior to the conclusion of the natural pulse set by the gain and energy extraction dynamics of the laser cavity. After opening the Q-switch to initiate pulse build-up, the Q-switch would be set back to a high loss state at some point, quickly reducing the circulating intensity and effectively clipping off the falling edge of the pulse. This could be implemented using many different types of Q-switches, including both electro-optic and acousto-optic varieties.
For low repetition-rate lasers where the pulse repetition frequency (PRF) is much lower than 1/(upper state lifetime), this pulse-clipping technique can shorten pulses effectively. The method has some efficiency cost compared to the free-running laser without pulse clipping, though, since stored energy is left behind in the gain medium and the long re-pumping times ensure that little or none of the energy left behind will be available for use by the following pulses.
At a PRF much higher than 1/(upper state lifetime), pulse-clipping is potentially much more advantageous, but also more complicated. Because the pulse is clipped by the Q-switch before the gain drops below the loss level, clipping off the falling edge of a pulse allows the residual net gain left behind after the shortened pulse to be seen by the following pulses, since the time between pulses is much less than the lifetime of the gain medium. For internally frequency converted lasers this could, in principle, be a significant advantage, since the low-nonlinear optical (NLO)-conversion-efficiency tail of the pulse would be clipped away and that stored energy saved for conversion at higher intensities and efficiencies in subsequent pulses.
In practice, however, this otherwise attractive scheme for internally frequency converted lasers typically allows only minimal pulse shortening and moderate improvements in efficiency before running into inherent stability limitations. As the Q-switch window (time between initiation of build-up and clipping of the pulse) decreases progressively and more gain is left behind after the pulse is clipped, a mechanism for communication between pulses develops which quickly destabilizes the pulse train. Thus, a solution to this gain-mediated instability problem is necessary for Q-switch pulse clipping to be a useful technique.
A simple model for the laser dynamics illustrates the stability problems that arise when pulse length is modified by clipping of the pulse trailing. This simple model is for a purely linear laser, i.e. no nonlinear output coupling, but serves to illustrate the stability issues which can occur in both linearly and nonlinearly output coupled lasers. Consider the two equations (1) and (2) for circulating power P and gain g in a linear laser in the high PRF limit. For the sake of simplification, we assume the pulse to be sufficiently short that re-pumping of the gain medium during the pulse can be ignored for determining pulse dynamics.
                                          T            RT                    ⁢                                    ⅆ              P                                      ⅆ              t                                      =                              (                          g              -              l                        )                    ⁢          P                                    Eq        .                                  ⁢                  (          1          )                                                              ⅆ            g                                ⅆ            t                          =                              -            gP                    /                      E            sat                                              Eq        .                                  ⁢                  (          2          )                    
The laser parameters included in this model are the cavity round trip time TRT, the cavity loss l, and an effective gain medium saturation energy, Esat, which depends on the saturation intensity of the gain medium and the cavity mode interaction with the gain medium (number of passes, mode size, etc.)
Fundamentally, a minimum condition for laser pulse train stability requires that a small perturbation (in gain, for example) on a particular pulse not be amplified in its impact on subsequent pulses. If such amplification occurs, the perturbation can eventually lead to undesirable behaviors such as period doubling. The laser described by the above-coupled equations above exhibits such instability if the circulating intensity is clipped during the pulse by an aggressively short Q-switch window. The same behavior occurs in internally frequency converted Q-switched lasers unless pains are taken to stabilize the pulse train as described below.
FIGS. 1a, 1b shows an example of prior art intracavity powers and gain levels as a function of time calculated by numerical solution of the coupled equations for two slightly different initial gain conditions. The intensity builds up at a rate set by the initial gain level, extracts energy from the gain medium, and finally decays as the gain level eventually falls below the cavity round trip loss level. The additional curve in FIG. 1 shows the difference in instantaneous gain at time T after the Q-switch is opened to initiate pulse build-up for the two initial conditions. The initial gain difference remains almost constant through the pulse build-up phase, but once significant energy extraction begins, a much larger difference in gain exists for most of the pulse duration. After the circulating intensity decays away, the instantaneous gain difference drops to a value lower than the initial gain difference at time t+0.
If we now suppose that a Q-switch were turned back to a high loss state at time T, we can infer the impact on stability by examining the gain differences existing at that time. If time T occurs significantly after the pulse intensity has decayed away, the gain difference at time T will always be less than or equal to the original gain level and the laser will be at least marginally stable, since any perturbation will decay away in time, however slowly. As soon as the time T begins to encroach on the pulse falling edge, though, the gain difference at time T can be significantly larger than the input gain difference. Any gain fluctuations will then be amplified in their effect on subsequent pulses. Defining a ratio of the gain difference at Q-switch closing time t=T to the initial gain difference at t=0 in the limit of small gain fluctuations, we see that pulse train stability with pulse clipping requires the minimum condition according to equation (3).
                                                                  ⅆ                              g                ⁡                                  (                  T                  )                                                                    ⅆ                              g                ⁡                                  (                  0                  )                                                                              <        1                            Eq        .                                  ⁢                  (          3          )                    
When the quantity in equation 3 is less than 1, any gain fluctuations will be damped in their effect in subsequent pulses; the smaller the quantity, the more quickly any fluctuations will damp away and the more stable the laser will be.
Examining the gain difference curve in FIG. 1c, the condition in equation (3) is met at times t=T much longer than the pulse, but is violated when the pulse is clipped significantly by the Q-switch. As a result, only minimal pulse shortening and control is possible using the Q-switch without pulse train destabilization.
This can be understood intuitively in a repetitively Q-switched laser the following way. Extra gain is available for a first pulse, which builds up more quickly and extracts more stored energy, leaving less gain available for the following pulse. Seeing less gain, that second pulse builds up more slowly and extracts less energy, leaving the right initial conditions for a yet larger first larger pulse.
FIGS. 2a and 2b show an example of the results of numerical simulations of the impact of the Q-switch clip time on pulse width in a prior art laser prone to this instability. As the Q-switch window (time between opening at time 0 and closure at time T) is reduced and begins to clip the failing edge of the pulse, a bifurcation in pulse energy (FIG. 2a) and pulse width (FIG. 2b) occurs, so that the pulse train contains alternating large and small pulses. As the window is closed yet further, the energy of the smaller pulse quickly goes to zero, so that the laser reaches threshold only on alternating Q-switch events and the repetition frequency is halved.
Such behavior can be observed in high repetition rate linearly and nonlinearly output coupled lasers when the Q-switch gate is reduced. In practice, pulse stability sets the practical lower bound on the width of the Q-switch window and prevents aggressive Q-switch clipping from being a useful technique for pulse width control.
The origin of the large gain differences and resulting instabilities is primarily timing delay between the pulses. The pulse intensity envelope and gain behavior are not so different for the two curves plotted in FIG. 1a-1c. If they could somehow be shifted to overlap properly in time, the stability condition in equation (3) could be met for clipping at an arbitrary time T. The timing of the pulses is determined primarily by the build-up time, which is inversely proportional to the gain available at the start of the pulse.
Ultimately, this stability problem arises because the energy extraction (during the pulse) occurs on a time scale which is short relative to the overall build-up time, so that small fluctuations in build-up time cause sufficient shifts in pulse timing to leave large gain discrepancies. If the build-up time were substantially shorter (i.e., if the circulating intensity were already macroscopic when the Q-switch opens), the stability against pulse clipping would be significantly enhanced, since any gain fluctuations would generate much smaller timing shifts.
However, achieving such a stable enhanced circulating intensity before opening the Q-switch is difficult given the proximity to the laser threshold intensity. Techniques such as pre-lasing and injection seeding can be used to raise the circulating intensity before a Q-switched pulse, but both have significant limitations (single frequency lasers, ring lasers, etc.).
It is worth noting that pulse timing is not the only source of pulse instability. Higher order transverse modes, if present, can also couple subsequent pulses and have been observed to cause similar period-doubling instabilities. Hence, the analysis here applies primarily to single transverse mode lasers.
The invention disclosed here takes a counterintuitive approach to pulse shortening. Increasing the nonlinear outcoupling in an internally frequency converted laser typically results in longer pulse lengths, all other things being equal, since the intensities are reduced and energy extraction from the gain medium is less rapid. However, increasing the nonlinear outcoupling also yields a significant benefit.
The additional nonlinear coupling in accordance with the invention disclosed here makes the pulse much more stable against reductions in the Q-switch gate width which would otherwise cause pulse instabilities. As a result of the increased non-linear outcoupling, the gate width can then be greatly reduced, to the point where the pulse length is determined primarily by the build-up time and Q-switch gate width rather than only by the dynamics of energy extraction from the gain medium. Since energy extraction dynamics no longer determine the pulse width, the pulse width can be reduced far below what is possible by optimizing the nonlinear output coupling and gain dynamics. This enables operation of intracavity frequency-converted lasers in an entirely new pulse width regime with much shorter pulses than are achievable through variation in the laser parameters alone.
The conditions in accordance with one aspect of the invention are now considered, under which a laser overcomes the previously presented stability problem, allowing pulse clipping and enabling the decoupling of the pulse width (equivalently pulse duration) from pulse energy and pulse repetition rate.
The key to this invention is the proper choice of the nonlinear coupling to achieve a residual level of gain in the gain medium, after the pulse is clipped at time T and the circulating intensity decays away, which is independent of the initial gain value present in the gain medium before the pulse build-up began. We define this point as the gain fluctuation insensitive nonlinear coupling level. Under ideal Gain Fluctuation Insensitivity Conditions, the quantity in equation (3) would be approximately zero for all values of time T after the pulse intensity peak, so that the pulse train would have maximum stability against behaviour such as period doubling, enabling aggressive pulse clipping without causing instability. As is clear from equation 3, however, stable pulse-clipped laser operation is possible over some range of nonlinear output coupling levels around the ideal gain independent condition as long as the absolute value of the quantity in equation 3 is less than 1. Therefore operation near the ideal gain fluctuation insensitive condition is sufficient to achieve the benefits accorded by this invention.
Since the analytical and numerical models presented below to explain and illustrate the invention and the required Gain Fluctuation Insensitive Condition are necessarily simplified for clarity, they do not capture all aspects of laser operation. For example, the simplified model does not include spatial gain saturation effects, using instead a constant effective saturation energy to describe the energy extraction behaviour of the pulse. Similarly, complicating effects in the frequency conversion process such as saturation of conversion efficiency at high intensities and high initial gains are not included. Pushed into the regime of large non-ideal effects, for example very high circulating intensities that predict greater than 100% conversion per round trip, the simple model will clearly be inadequate for prediction of gain fluctuation insensitive behaviour.
However, we demonstrate that gain fluctuation insensitive nonlinear output coupling conditions are clearly identifiable by numerical calculation over a wide and useful range of initial gain values, where more complicated effects do not dominate the laser pulse behaviour. Actually, given that stable operation is achievable over a range around the ideal gain independent condition, the benefits of operating near the gain independent condition will be obtainable to some degree even in the regime where effects beyond those included in the simple model become significant. For this reason, it is also possible that a single laser could achieve the benefits of gain fluctuation insensitive nonlinear coupling at some pulse repetition rates, while not achieving it at others due to these kinds of additional effects.
The pulse behavior during the build-up and the decay phases can be described analytically for the case of second harmonic generation, which allows one skilled in the art to estimate the conditions necessary for gain insensitivity and pulse train stability. Subsequent numerical analysis will lay out that condition more precisely.
To estimate analytically the Gain Fluctuation Insensitivity Condition for a nonlinearly coupled second harmonic generation laser, the same parameters are used as above except for the addition of an output coupling which is proportional to the square of the circulating power (second harmonic generation). Since the output of this laser is at the second harmonic and the linear cavity losses can be minimized as is known in the art, the linear losses will be neglected. Thus, equations 1 and 2 can be rewritten as equations 4 and 5 respectively. Equation 4 includes α, which determines the magnitude of the nonlinear output coupling by second harmonic generation:
                                          T            RT                    ⁢                                    ⅆ              P                                      ⅆ              t                                      =                  gP          -                      α            ⁢                                                  ⁢                          P              2                                                          Eq        .                                  ⁢                  (          4          )                                                              ⅆ            g                                ⅆ            t                          =                  -                      gP                          E              sat                                                          Eq        .                                  ⁢                  (          5          )                    
The second-order nonlinear system defined in equations (4) and (5) has a one-parameter family of solutions which can be written in closed form as expressed in equations (6, 7).
                                          P            ⁡                          (              t              )                                =                                    E              sat                                      t              -                              T                0                                                    ,                              g            ⁡                          (              t              )                                =                                                                      α                  ⁢                                                                          ⁢                                      E                    sat                                                  -                                  T                  RT                                                            t                -                                  T                  0                                                      .                                              Eq        .                                  ⁢                  (                      6            ,            7                    )                    
These exact solutions are not physical, in that the power and the gain increase without bound as the time goes back to the integration constant T0. However, this special family is asymptotically close to the decay behavior of the desired solutions if the SHG coupling a is large compared with TRT/Esat.
During the decay, the gain is less than the nonlinear loss, so that g<αP. The special family has g=(αTRT/Esat)P, and thus the solutions satisfy (αTRT/Esat)P<g<αP during the decay. If α>>TRT/Esat, then the solutions are closely bounded and are nearly equal to the special solutions during the decay period. Two initial conditions apply: a small noise power Pinitial and some finite gain, gi, which will determine the value of T0 for the correct asymptotic decay curve.
During the build-up of the pulse, the power increases exponentially, as expressed in equation (8):P(t)=Pinitalexp(git/TRT),   Eq. (8)
The gain is thus nearly constant.
In between the build-up and the decay is the peak part of the pulse—the last part of the rise and first part of the decay—where all terms in the differential equations are important and no general formulas are available. Fortunately, the duration of this peak is short and relatively little gain is depleted during this part of the pulse evolution. Thus, to a simplest approximation, the build-up and decay behavior can be matched directly, as expressed in equation (9):
                                          t            p                    =                                                    T                RT                                            g                i                                      ⁢                          ln              ⁡                              (                                                      P                    p                                    /                                      P                    initial                                                  )                                                    ,                            Eq        .                                  ⁢                  (          9          )                    
where Pp is the peak power. From the decay behavior, the time of the peak is approximately
                                          t            p                    =                                                                      α                  ⁢                                                                          ⁢                                      E                    sat                                                  -                                  T                  RT                                                            g                i                                      +                          T              0                                      ,                            Eq        .                                  ⁢                  (          10          )                    
where T0 is again the free parameter of the special family.
In general, this connection (the two expressions for tp) will imply the dependence of T0 on gi, Pinitial, and on the laser parameters α, Esat, and TRT. Now the strong condition is imposed that T0 be independent of the initial gain gi, so that all pulses decay along the same curve independent of the initial gain value before the pulse. This ensures that the final gain behavior is independent of the initial gain and there can be no communication or interaction between pulses. In that case T0 must be identically zero, which implies that
                              α          gfi                ≈                                                            T                RT                                            E                sat                                      ⁡                          [                                                ln                  ⁡                                      (                                                                  P                        p                                            /                                              P                        initial                                                              )                                                  +                1                            ]                                .                                    Eq        .                                  ⁢                  (          11          )                    
The nonlinear output coupling value for achieving the Gain Fluctuation Insensitive Condition, αgfi, depends on the initial intensity circulating in the cavity before the Q-switch opens, with higher nonlinear coupling required for lower initial intensity levels. Since the pre-pulse circulating power in almost all Q-switched lasers is many orders of magnitude below the peak power, the factor in square brackets is much larger than unity, and the term “+1” can be neglected.
In the analysis of nonlinearly-output-coupled peak power, pulse width, and efficiency, Murray and Harris determined an optimal coupling point α,α≈TRT/Esat (β+1 in the original paper),  Eq. (12)
at which the highest harmonic peak power was achieved, the efficiency was close to the maximum, and the pulse width had increased only a small fraction over the minimum value. As the nonlinear output coupling is increased beyond the optimal value determined by Murray and Harris, the pulse width will only lengthen further. For nonlinear coupling values lower than this value, the pulse width of a nonlinear-output-coupled second harmonic generation laser is approximately constant at value we define here as the Characteristic Minimum Pulse Width achievable by a laser having the same parameters (other than the nonlinear coupling, which is being varied).
Compared to the nonlinear coupling value optimized for harmonic peak power as disclosed in the prior art of Murray and Harris, it can be seen that the Gain Fluctuation Insensitive Condition will typically require many times (a factor of the order of 20) greater nonlinear output coupling.
As a result of this unusually large nonlinear coupling required to achieve the Gain Fluctuation Insensitive Condition, the free-running pulse width (without pulse clipping) of laser disclosed herein will be much longer than in the optimally nonlinearly-coupled case. However, because of the enhanced pulse train stability, the disclosed laser will now be stable against Q-switch clipping of the pulse falling edge and dramatic shortening of the output pulse widths will be possible. Thus, the laser operating at the Gain Fluctuation Insensitive Condition and using pulse-clipping will be able to generate output harmonic wavelength pulses much shorter than the Characteristic Minimum Pulse Width identified by Murray and Harris.
More detailed investigation into the conditions for maximal pulse train stability requires numerical simulation of the coupled equations to connect the build-up to the pulse decay through the region of peak nonlinear conversion. For the simulations of another embodiment Esat is taken to be 3 mJ, TRT is taken to be 5 ns, and the initial noise input is taken to be 10 μWatt to model approximately the experimental results presented in the following section.
FIG. 3 shows the circulating infrared power intensity and gain as a function of time for several input gain levels while applying the “optimal” nonlinear coupling level of α=0.00166/kW for maximum harmonic peak power as determined by Murray and Harris in prior art. For these conditions, the nonlinear coupling level is far short of the value required for gain fluctuation insensitivity.
Pulses with higher initial gain levels reach their peak intensity and decay earlier in time, leading to large post-pulse gain differences and pulse train instability if the pulse were clipped shortly after the intensity peak.
Levels of nonlinear coupling near the gain fluctuation insensitive condition α=0.035/kW are used in accordance with this invention. This level is determined empirically by observing the overlay of the curves in time. For comparison, using typical values of Pp=10 kW and Pinitial=10 μWatt, equation (11) predicts α=0.033/kW. 4a to 4c show the numerical simulation results for a laser generating intracavity second harmonic output in accordance with this invention, with graphs of the circulating intensity and gain as a function of time for several initial gain levels, for the nonlinear coupling values, α, of 0.0175/kW (4a), 0.035/kW (4b, the gain fluctuation insensitivity condition) and 0.070/kW (4c).
As expected at the Gain Fluctuation Insensitivity Condition shown in FIG. 4b, the individual gain curves all fall onto a single universal curve for times slightly after the intensity peaks, despite the substantial difference (factors of about 5 times) in initial gain values shown. The intensity curves also fall onto a similar universal decay curve at a time shortly afterward. It is important to note that the same basic behavior occurs if linear losses and re-pumping are included, although at low initial gain values some deviation from the ideal curve is observed due to the greater relative importance of the linear losses. FIGS. 4a and 4c, in which the nonlinear coupling is decreased and increased by a factor of 2 from the Gain Fluctuation Insensitivity Condition, illustrate that stability can be achieved over some range around the ideal condition, as is indicated by equation (3). No large differences in gain appear after the pulse intensity peaks, unlike in the more typical prior art situation illustrated in FIG. 3.
When the pulse falling edge is clipped by the Q-switch and significant gain is left behind after the pulse, the background gain level will rise. As this gain level rises, the build-up time will shorten, the peak intensity will increase, and the efficiency of conversion to the harmonic will increase as a result. A new equilibrium will be reached when the power extracted by the nonlinear conversion during the Q-switch open period increases to equal (modulo any additional power lost from linear cavity losses) the available power/pulse at the relevant repetition rate.
This decoupling of background gain level from individual pulse energy has significant implications for laser operation. For a laser without pulse clipping, the build-up, peak intensity, and nonlinear conversion efficiency are set by the energy (gain) available for use by that particular pulse, since all net gain is extracted by each pulse. As the repetition rate of such an internally frequency converted laser increases, the output power at the harmonic will typically drop.
In contrast, a pulse-clipped laser with increased background gain level could in principle keep the same pre-pulse gain level as the PRF increases, preserving the peak intensity, nonlinear conversion efficiency, and overall laser output power at the harmonic wavelength. Hence, clipped-pulse operation should enable significant PRF independence, particularly at very high PRF values.
There are obviously some constraints on the effectiveness of this technique, as the periodic cavity dumping of circulating intensity throws energy away. This loss increases with repetition rate, since the cavity dumping occurs more frequently, and with increasing circulating intensity, since more power will be thrown away with each cavity dumping event.
Another intuitive way of understanding this Gain Fluctuation Insensitivity Condition is to see that each pulse automatically removes exactly the right amount of energy to leave behind the same residual gain regardless of Q-switch window width. Therefore, any perturbations are removed by a single pulse and have no impact on subsequent pulses.
The case of gain fluctuation insensitive nonlinear coupling for third harmonic generation is not so amenable to analytical analysis, and so must be investigated by numerical methods. We model a laser with similar parameters to the previous numerical example, but now include both second harmonic and third harmonic generation terms, so that equation 4 must be replaced by equation 13 to account for the additional nonlinear conversion stage. Note that all radiation generated at second and third harmonic wavelengths is output coupled from the laser cavity.
                                                        T              RT                        ⁢                                          ⅆ                P                                            ⅆ                t                                              =                      gP            -                          α              ⁢                                                          ⁢                              P                2                                      -                          α              ⁢                                                          ⁢                              P                2                            ⁢                                                sin                  2                                ⁡                                  (                                                            β                      ⁢                                                                                          ⁢                      P                                                        )                                                                    ,                            Eq        .                                  ⁢        13            
In equation 13, α is again the second harmonic nonlinear output coupling and β now sets the level of coupling from the fundamental and second harmonic to the third harmonic. For this simple model of third harmonic generation, a Gain Fluctuation Insensitive level of nonlinear output coupling can be clearly identified by numerical simulation, as is illustrated in FIGS. 8a, b, and c. The gain vs. time curves in FIG. 8b clearly show that a gain fluctuation insensitive nonlinear output coupling has been achieved, as the gain curves join the “universal” gain decay curve shortly after the fundamental intensity peak, just as was observed for the second harmonic generation case investigated previously. In this case there is no single parameter for the Gain Fluctuation Insensitive Condition, since the second harmonic coupling coefficient and the third harmonic coupling coefficient both affect the gain behaviour. For the Gain Fluctuation Insensitive Condition example shown in FIG. 8b, α=0.0175/kW, while β=0.2/kW. FIGS. 8a and 8c show the same laser with the alpha decreased (α=0.00875) and increased (α=0.035) by a factor of 2. Gain fluctuation insensitive operation is therefore achievable in the case of nonlinear output coupling by third harmonic generation and will allow pulse width reduction by aggressive Q-switch pulse clipping in a similar manner.
It should be realized that lasers enabled by this invention can be used in many different applications and modified in many different ways while remaining within the scope of the invention. Lasers enabled by this invention can be used for material processing, scientific, medical, remote sensing, and security applications. Specific parts of the laser can be modified such as the type of gain medium such as solid-state, liquid, or ion and the number of separate gain media used in the laser. The types of nonlinear materials or the number of separate nonlinear materials or crystals used to accomplish nonlinear frequency conversion can change. External frequency conversion of the enabled laser output beam after output coupling from the laser could be used to further change the frequency of the laser radiation. The method of pumping could change to include laser diodes, lamps, or electrical discharge. Both single and multi-transverse lasers and single or multi-longitudinal modes laser could be built using this invention. Finally, seeding techniques could be used in conjunction with this invention to modify pulse buildup times and gain-independent conditions.